Related papers: A uniqueness theorem for stable homotopy theory

We present a method for computing $\mathbb{A}^1$-homotopy invariants of singularity categories of rings admitting suitable gradings. Using this we describe any such invariant, e.g. homotopy K-theory, for the stable categories of…

The notion of a natural model of type theory is defined in terms of that of a representable natural transfomation of presheaves. It is shown that such models agree exactly with the concept of a category with families in the sense of Dybjer,…

In this article we consider the homotopy theory of stratified spaces through a simplicial point of view. We first consider a model category of filtered simplicial sets over some fixed poset $P$, and show that it is a simplicial…

This note extends Quillen's Theorem A to a large class of categories internal to topological spaces. This allows us to show that under a mild condition a fully faithful and essentially surjective functor between such topological categories…

We recapture Douglas' framework for twisted parametrized stable homotopy theory in the language of $\infty$- categories. A twisted spectrum is essentially a section of a bundle of presentable stable $\infty$-categories whose fiber is the…

We discuss the current state of knowledge of stable homotopy groups of spheres. We describe a new computational method that yields a streamlined computation of the first 61 stable homotopy groups, and gives new information about the stable…

A separable, proper morphism of varieties with geometrically connected fibers induces a homotopy exact sequence relating the \'etale fundamental groups of source, target and fiber. Extending work of dos Santos, we prove the existence of an…

We give a homotopy classification of the global defects in ordered media, and explain it via the example of biaxial nematic liquid crystals, i.e., systems where the order parameter space is the quotient of the $3$-sphere $S^3$ by the…

We construct a spectral sequence associated to a stratified space, which computes the compactly supported cohomology groups of an open stratum in terms of the compactly supported cohomology groups of closed strata and the reduced cohomology…

Let $G$ be a finite group. Let $U_1,U_2,\dots$ be a sequence of orthogonal representations in which any irreducible representation of $\oplus_{n \geq 1} U_n$ has infinite multiplicity. Let $V_n=\oplus_{i=1}^n U_n$ and $S(V_n)$ denote the…

This paper studies the foundations of the geometric fixed point functor in multiplicative equivariant stable homotopy theory. We introduce a new class of equivariant orthogonal spectra called generalized orbit desuspension spectra and…

Building on the work of Martin Stolz, we develop the basics of equivariant stable homotopy theory starting from the simple idea that a G- spectrum should just be a spectrum with an action of G on it, in contrast to the usual approach in…

In this note we explain that homotopy coherent simplicial nerve has to used intead of the standard definition in the author's papers on formal deformation theory. A convenient version of the notion of fibered category is presented which is…

This is an introductory textbook to univalent mathematics and homotopy type theory, a mathematical foundation that takes advantage of the structural nature of mathematical definitions and constructions. It is common in mathematical practice…

Let $R$ be a right notherian ring. We introduce the concept of relative singularity category $\Delta_{\mathcal{X}}(R)$ of $R$ with respect to a contravariantly finite subcategory $\mathcal{X}$ of $\rm{mod}\mbox{-}R.$ Along with some…

Having spectral correlations that, over small enough energy scales, are described by random matrix theory is regarded as the most general defining feature of quantum chaotic systems as it applies in the many-body setting and away from any…

In homotopy type theory (HoTT), all constructions are necessarily stable under homotopy equivalence. This has shortcomings: for example, it is believed that it is impossible to define a type of semi-simplicial types. More generally, it is…

A stratified space is a topological space together with a decomposition into strata corresponding to different types of singularities. Examples of such spaces appear everywhere in topology and geometry. The study of stratified spaces…

We survey several mathematical developments in the holonomy approach to gauge theory. A cornerstone of this approach is the introduction of group structures on spaces of based loops on a smooth manifold, relying on certain homotopy…

We construct a Quillen model structure on the category of spectral categories, where the weak equivalences are the symmetric spectra analogue of the notion of equivalence of categories.